given $G:[0,\infty)\rightarrow \mathbb R$
$$ G'(t)\geq -\lambda G(t)+ f(G(t)) $$
where $\lambda >0$ and $f$ is a convex function. $G(0)>0$ is also given.
Show that $G$ blows up at finite time $t>0$.
I can see that in case of $f(x)=x^p$ for $p>1$. Please help me in generalizing it to any convex function.
First of all, $G(0)>0$ is not enough to guarantee finite time blow up. If $G(u)=u^p$, $p>1$, you need $G(0)>\lambda^{1/(p-1)}$.
Secondly, convexity does not guarantee blow up in finite time. Consider $G(u)=u\log(1+u)$. It is convex, but all solutions are global.